Regular vertex diameter critical graphs

Abstract

A graph is called vertex diameter critical if its diameter increases when any vertex is removed. Regular vertex diameter critical graphs of every valency k ≥ 2 and diameter d ≥ 2 exist, raising the question of identifying the smallest such graphs. We describe an infinite family of k-regular vertex diameter critical graphs of diameter d with at most kd+ (2k − 3) vertices. This improves the previously known upper bound for all odd valencies k.